Quark mass dependence of $\pi \pi$ scattering in isospin 0, 1, and 2 from lattice QCD

Using lattice QCD we extract $\pi\pi$ scattering amplitudes with isospin--0,1,2 in low partial-waves at two values of the light quark mass corresponding to $m_\pi \sim 283$ and $330$ MeV. We confirm expectations of weak repulsion in isospin--2, and the presence of a narrow $\rho$ resonance in isospin--1, and study the pion mass dependence of these channels. In isospin--0 we find that the two pion masses considered straddle the point at which the $\sigma$ transitions from being a stable bound-state to being either a virtual bound-state or a subthreshold resonance. We discuss the ability of lattice calculations like these to precisely determine the $\sigma$ pole location when it is a resonance, and propose an approach in which the full complement of amplitudes computed in this paper can be used simultaneously to provide more constraint.